It is apparent that a generic Riemannian manifold does not carry any umbilical submanifold of dimension \(n\ge 2\) . For instance, it was shown by Souam and Toubiana [1]} that of the eight three-dimensional model geometries of Thurston, only the spaces of constant curvature, the product spaces \(\mathbb {S}^2\times \mathbb {R}\)  and \(\mathbb {H}^2\times \mathbb {R}\) , and the \(\emph {Sol}\)  geometry admit umbilical surfaces, in the latter case a unique non totally geodesic one, up to ambient isometries. Here \(\mathbb {S}^n\)  and \(\mathbb {H}^n\)  stand for the sphere and hyperbolic space of dimension \(n\) , respectively. On the other hand, by a result of Leung and Nomizu [2]} (see also Section \(1.7\)  of [3]}), if \(N\)  is a Riemannian manifold that satisfies the axiom of \(r\) -spheres, that is, for any point \(x\in N\)  and for any integer \(r\ge 2\) , there always exists an extrinsic sphere of \(N\)  tangent at \(x\)  to any prescribed \(r\) -dimensional subspace \(V\subset T_xN\) , then \(N\)  must have constant sectional curvature. Recall that an extrinsic sphere of a Riemannian manifold \(N\)  is an umbilical submanifold of \(N\)  whose mean curvature vector field is parallel in the normal connection. It is also pointed out in [2]} that if the requirement on the mean curvature vector field is dropped in the axiom of \(r\) -spheres, then the corresponding weaker axiom for \(n > 4\)  and \(3 < r < n\)  implies that M is conformally flat.
